A general iterative algorithm for equilibrium problems and strict pseudo-contractions in Hilbert spaces

A general iterative algorithm for equilibrium problems and strict pseudo-contractions in Hilbert spaces MING TIAN College of Science Civil Aviation University of China Tianjin 300300, China P. R. CHINA tianming1963@126.com LEI LIU College of Science Civil Aviation University of China Tianjin 300300, China P. R. CHINA leolei2010@gmail.com Abstract: In this paper an iterative scheme is presented for finding a common element of the set of solutions of the variational inequality, fixed points of strict pseudo-contraction and solutions of equilibrium problem in Hilbert spaces. Under suitable conditions, it is proved that implicit and explicit schemes are of strong convergence properties. Obtained results improve and extend the existed results. Key Words: Nonexpansive mapping, Fixed point, Equilibrium problem, Strict pseudo-contraction, Variational inequality, Iterative algorithm 1 Introduction Throughout this paper, we always assume that H is a real Hilbert space with inner product, and norm. Let C be a nonempty closed convex subset of H. Recall that a mapping T from C into itself is nonexpansive if T x T y x y, x, y C. The set of fixed points of T is denoted by F (T ), that is, F (T ) = {x D(T ) : T x = x}; If C H is nonempty, bounded, closed and convex and T is a nonexpansive self-mapping on C, then F (T ) is nonempty. Let A : C C be an operator, the variational inequality problem is to find x C such that V I : Ax, x x 0, x C. (1) The set of solutions of (1) is denoted by V I(C, A). Multiple iterative schemes have been proposed to approximating the fixed point of an operator, which also is a solution of the variational inequality. In 1953, Mann [1] proposed the following iterative scheme: x n+1 = α n x n + (1 α n )T x n, (2) where x 0 C is an initial guess arbitrarily. If α n [0, 1] satisfying n=0 α n(1 α n ) =, then the sequence {x n } generated by (2) converges weakly to a fixed point of T. In 2000, Moudafi [3] introduced the viscosity approximation method for nonexpansive mappings. Let f be a contraction on H, starting with an arbitrary initial x 0 H, define a sequence {x n } recursively by x n+1 = α n f(x n ) + (1 α n )T x n, n 0, (3) where {α n } is a sequence in (0, 1). Xu [4] proved that under certain appropriate conditions on {α n }, the sequence {x n } generated by (3) strongly converges to the unique solution x C of the variational inequality (I f)x, x x 0, x F (T ). In 2001, Yamada [5] introduced the following hybrid iterative method: x n+1 = T x n µ F (T x n ), n 0, (4) where F is a k-lipschitzian and η-strongly monotone operator with k > 0, η > 0, 0 < µ < 2η/k 2, and then he proved that if { } satisfies appropriate conditions, the sequence {x n } generated by (4) converges strongly to the unique solution x F (T ) of the variational inequality F x, x x 0, x F (T ). In 2006, Marino and Xu [6] introduced the general iterative algorithm x n+1 = α n γf(x n ) + (I α n A)T x n, n 0, (5) where T is a self-nonexpansive mapping on H, f is a contraction of H into itself with coefficient ρ (0, 1) ISSN: 1109-2769 480 Issue 12, Volume 10, December 2011

satisfies certain condition, and A is a strongly positive bounded linear operator on H. He proved that {x n } generated by (5) converges strongly to a fixed point x of T, which is the unique solution to the following variational inequality: (γf A)x, x x 0, x F (T ), and is also the optimality condition for some minimization problem. In 2010, M. Tian [7] introduced the general iterative algorithm x n+1 = α n γf(x n ) + (I µα n F )T x n, n 0, (6) where F : H H is an L-Lipschitzian and η- strongly monotone operator with L, η > 0. Under some mild assumptions, he proved that {x n } generated by (6) converges strongly to a point x F (T ), which is also the unique solution of the following variational inequality: (µf γf)x, x x 0, x F (T ). Recently, many authors considered the iterative schemes to approximating the fixed point of a strictly pseudo-contraction. A mapping S : C H is said to be k-strictly pseudo-contractive if there exists a constant k [0, 1) such that Sx Sy 2 x y 2 + k (I S)x (I S)y 2, x, y C. Note that the class of k-strict pseudo-contraction strictly includes the class of nonexpansive mapping, that is, S is nonexpansive if and only if S is 0- srtict pseudo-contractive; it is also said to be pseudocontractive if k = 1. Clearly, the class of k-strict pseudo-contractions falls into the one between classes of nonexoansive mappings and pseudo-contractions. For finding an element of F (S) V I(C, A), Takahashi and Toyoda [8] introduced the following iterative scheme: x 1 C and x n+1 = α n x n + (1 α n )SP C (x n Ax n ), n 1, and obtained a weak convergence theorem in a Hilbert space, where {α n } and { } are sequences satisfied certain conditions. For finding an element of F (T ), Qin et al [9] introduce a composite iterative scheme as follows with x 1 = x H: { yn = P K [β n x n + (1 β n )T x n ], x n+1 = α n γf(x n ) + (I α n A)y n, n 1, where T is a non-self k-strict pseudo-contraction, f is a contraction and A is a strong positive linear bounded operator. Under certain appropriate assumptions on the sequence {α n } and {β n }, the iterative sequence {x n } converges strongly to a fixed point of the k-strict pseudo-contraction, which also solves some variational inequality. Let ϕ be a bifunction of C C into R. The classical equilibrium problem for ϕ is to find x C such that EP : ϕ(x, y) 0, y C, (7) denoted the set of solutions by EP (ϕ). Given a mapping T : C H, let ϕ(x, y) = T x, y x, x, y C, then z EP (ϕ) if and only if T z, y z 0 for all y C, that is, z is a solution of the variational inequality. Numerous problems in physics, optimizations and economics reduce to find a solution of (7). Some methods have been proposed to solve the equilibrium problem, see for instance, [2, 3] and the references therein. By using the general approximation method, many authors constructed the compositive schemes to obtained the common element of fixed points of nonexpansive mapping and solutions of equilibrium problem. Next, we list their main contributions. For finding an element of EP (ϕ) F (S), Ceng et al [10] established the following iterative scheme: { ϕ(un, y) + 1 r n x n+1 = α n u n + (1 α n )Su n, n N, under certain conditions, the sequences {x n } and {u n } converge weakly to an element of EP (ϕ) F (S). Under the same conditions, the sequences {x n } and {u n } converge strongly to an element of EP (ϕ) F (S) if and only if lim inf d(x n, EP (ϕ) F (S)) = 0, where d(x n, EP (ϕ) F (S)) denotes the metric distance from the point x n to EP (ϕ) F (S). To find an element of EP (ϕ) F (S), Takahashi and Takahashi [11] introduced the following iterative scheme by the viscosity approximation method in a Hilbert space: x 1 H and { ϕ(un, y) + 1 r n x n+1 = α n f(x n ) + (1 α n )Su n, n N. Under suitable conditions, some strong convergence theorems are obtained. ISSN: 1109-2769 481 Issue 12, Volume 10, December 2011

For finding an element of EP (ϕ) V I(C, A) F (S), recently Su, Shang and Qin [12] introduced the following iterative scheme: x 1 H and r n x n+1 = α n f(x n ) + (1 α n )SP C (u n Au n ), n N. Under suitable conditions, some strong convergence theorems are obtained, which are extend and improve the results of Iiduka et al [13] and Takahashi et al [11]. To find an element of EP (ϕ) V I(C, A) F (S), Plubtieng and Punpaeng [14] also introduced the following iterative scheme: x 1 = u C and r n y n = P C (u n Au n ), x n+1 = α n u + β n x n + γ n SP C (y n Ay n ), n N. Under suitable conditions, some strong convergence theorems are obtained, which are extend some recent result of Yao and Yao [15]. In 2009, Y. Liu [16] introduced two iterative schemes by the general iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a k-strictly pseudo-contractive non-self mapping in the setting of a real Hilbert space. y n = β n u n + (1 β n )Su n, x n = α n γf(x n ) + (I α n B)y n, n N, and x 1 H arbitrarily, y n = β n u n + (1 β n )Su n, x n+1 = α n γf(x n ) + (I α n B)y n, n N where B is a strong positive bounded linear operator on H. Under some assumptions, the strong convergence theorems are obtained. In 2011, M. Tian [17] adopted the hybrid steepest descent methods for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudo-contraction mapping in the setting of real Hilbert spaces. y n = β n u n + (1 β n )Su n, x n = (I α n µf )y n, n N, and x 1 H arbitrarily, y n = β n u n + (1 β n )Su n, x n+1 = (I α n µf )y n, n N, where F : H H be an L-Lipschitzian continuous and η-strongly monotone operator on H with L, η > 0. Under some assumptions, the strong convergence theorems are obtained. Motivated and inspired by these facts, in this paper, we introduce two iteration methods, for finding an element of EP (ϕ) F (S), where S : C H is a k- strictly pseudo-contractive non-self mapping, which is also a solution of a variational inequality and then obtained two strong convergence theorems. Our results include Plubtieng and Punpaeng [18], S. Takahashi and W.Takahashi [11], Tada and W.Takahashi [19], Y. Liu [16] and M. Tian [17] as special cases. Furthermore, this paper will also be the development of the results of Ceng et al [10] in different directions. 2 Preliminaries Throughout this paper, we write x n x to indicate that the sequence {x n } converges weakly to x and x n x implies that {x n } converges strongly to x. For any x H, there exists a unique nearest point in C, denoted by P C x, such that x P C x x y, y C Such a P C x is called the metric projection of H onto C. It is known that P C is nonexpansive. Furthermore, for x H and u C, u = P C x x u, u y 0, y C. A mapping F : C C is called L-Lipschitzian if there exists a positive constant L, such that F x F y L x y, x, y C. Obviously, F is nonexpansive if and only if L = 1. F is said to be η-strongly monotone if there exists a positive constant η such that F x F y, x y η x y, x, y C. It is wildly known that H satisfies Opial s condition [13], that is, for any sequence {x n } with x n x, the inequality lim inf x n x < lim inf x n y, holds for every y H with y x. In order to solve the equilibrium problem for a bifunction ϕ : C C R, let us assume that ϕ satisfies the following conditions: (A1) ϕ(x, x) = 0, for all x C; ISSN: 1109-2769 482 Issue 12, Volume 10, December 2011

(A2) ϕ is monotone, that is, ϕ(x, y)+ϕ(y, x) 0, for all x, y C; (A3) For all x, y, z C, lim t 0 ϕ(tz + (1 t)x, y) ϕ(x, y); (A4) For each fixed x C, the function y ϕ(x, y) is convex and lower semicontinuous. Let us recall the following lemmas which will be useful for our paper. Lemma 1 (see [20]).Let ϕ be a bifunction from C C into R which satisfying (A1), (A2), (A3) and (A4), then, for any r > 0 and x H, there exists a point z C such that Further, if T r x = ϕ(z, y) + 1 y z, z x 0, y C. r { z C : ϕ(z, y) + 1 r then the followings hold: (1) T r is single-valued; (2) T r is firmly nonexpansive, that is, y z, z x 0, y C }, T r x T r y 2 T r x T r y, x y, x, y H; (3) F (T r ) = EP (ϕ); (9) EP (ϕ) is nonempty, closed and convex. Lemma 2 (see [21]).If S : C H is a k-strict pseudo-contraction, then the fixed point set F (S) is closed convex, so that the projection P F (S) is well defined. Lemma 3 (see [22]).Let S : C H be a k-strict pseudo-contraction. Define T : C H by T x = λx + (1 λ)sx, for each x C, then as λ [k, 1), T is a nonexpansive mapping such that F (T ) = F (S). Lemma 4 (see [23]).In a Hilbert space H, there holds the inequality x + y 2 x 2 + 2 y, x + y, x, y H. Lemma 5 (see [4]).Let {s n } be a sequence of nonnegative numbers satisfying the condition s n+1 (1 γ n )s n + γ n δ n, n 0, where {γ n }, {δ n } are sequences of real numbers such that: (i){γ n } [0, 1] and n=0 γ n =, (ii) lim sup δ n 0 or n=0 γ n δ n <. Then, lim s n = 0. Lemma 6 (see [7]).Let H be a Hilbert space, f : H H is a contraction with coefficient 0 < ρ < 1, and F : H H is an L-Lipschitz continuous and η- strongly monotone operator with L > 0, η > 0. Then for 0 < γ < µη/ρ, x y, (µf γf)x (µf γf)y (µη γρ) x y 2, x, y H. That is, (µf γf) is strongly monotone operator with coefficient µη γρ. 3 Main Results In the rest of this paper we always assume that F is an L-Lipschitzian continuous and η-strongly monotone operator with L, η >0 and assume that 0 < γ < τ ρ, τ = µ(η µl 2 /2). Let {T λn } be mappings defined as Lemma 1. Define a mapping S n : C H by S n x = β n x + (1 β n )Sx, x C, where β n [k, 1), then, by Lemma 3, S n is a nonexpansive. We consider the mapping G n on H defined by G n x = [I α n (µf γf)]s n T λn x, x H, n N, where α n (0, 1). By Lemma 1 and 3, we have G n x G n y [1 α n (τ γρ)] T λn x T λn y [1 α n (τ γρ)] x y. It is easy to see that G n is a contraction. Therefore, by the Banach contraction principle, G n has a unique fixed point x F n H such that x F n = [I α n (µf γf)]s n T λn x F n. For simplicity, we will write x n for x F n provided no confusion occurs. Next, we prove that the sequence {x n } converges strongly to a point q F (S) EP (ϕ) which solves the variational inequality (µf γf)q, p q 0, p F (S) EP (ϕ). (8) Equivalently, q = P F (S) EP (ϕ) [I (µf γf)]q. ISSN: 1109-2769 483 Issue 12, Volume 10, December 2011

Theorem 7 Let C be a nonempty closed convex subset of a real Hilbert space H and ϕ be a bifunction from C C into R satisfying (A1), (A2), (A3) and (A4). Let S : C H be a k-strictly pseudo-contractive nonself mapping such that F (S) EP (ϕ). Let F : H H be an L-Lipschitzian continuous and η- strongly monotone operator on H with L, η > 0 and 0 < γ < τ ρ, τ = µ(η µl2 /2). Let {x n } be a sequence generated by y n = β n u n + (1 β n )Su n, x n = [I α n (µf γf)]y n, n N, where u n = T λn x n, y n = S n u n, and { } (0, + ) satisfy lim inf > 0, if {α n } and {β n } satisfy the following conditions: (i) {α n } (0, 1), lim α n =0; (ii) 0 k β n λ < 1 and lim β n = λ, then {x n } converges strongly to a point q F (S) EP (ϕ) which solves the variational inequality (8). Proof: First, take p F (S) EP (ϕ). Since u n = T λn x n and p = T λn p, from Lemma 1, for any n N, we have u n p = T λn x n T λn p x n p. (9) Then, since y n = S n u n and S n p = p, we obtain that y n p = S n u n S n p u n p x n p. (10) Further, we have x n p = [I α n (µf γf)]y n p = (I α n µf )y n (I α n µf )p +α n γf(y n ) α n γf(p) +α n γf(p) α n µf p (1 α n τ) y n p + α n γρ y n p +α n (γf µf )p (1 α n τ) x n p + α n γρ x n p +α n (γf µf )p [1 α n (τ γρ)] x n p +α n (γf µf )p. It follows that x n p 1 τ γρ (γf µf )p. Hence, {x n } is bounded, and we also obtain that {u n } and {y n } are bounded. Notice that u n y n u n x n + x n y n = u n x n + [I α n (µf γf)]y n y n = u n x n + α n (µf γf)y n. (11) By Lemma 1, we have It follows that u n p 2 T λn x n T λn p 2 x n p, u n p = 1 2 ( x n p 2 + u n p 2 u n x n 2 ). u n p 2 x n p 2 u n x n 2. (12) Thus, from Lemma 4, (10) and (12), we obtain x n p 2 = [I α n (µf γf)]y n p 2 = [I α n (µf γf)]y n [I α n (µf γf)]p α n (µf γf)p 2 [I α n (µf γf)]y n [I α n (µf γf)]p 2 +2α n (µf γf)p, x n p [1 α n (τ γρ)] 2 y n p 2 +2α n (µf γf)p x n p [1 α n (τ γρ)] 2 u n p 2 +2α n (µf γf)p x n p [1 α n (τ γρ)] 2 ( x n p 2 x n u n 2 ) +2α n (µf γf)p x n p = [1 2α n (τ γρ) + αn(τ 2 γρ) 2 ] x n p 2 [1 α n (τ γρ)] 2 x n u n 2 + 2α n (µf γf)p x n p [1 + αn(τ 2 γρ) 2 ] x n p 2 [1 α n (τ γρ)] 2 x n u n 2 +2α n (µf γf)p x n p. It follows that [1 α n (τ γρ)] 2 x n u n 2 α 2 n(τ γρ) 2 ] x n p 2 +2α n (µf γf)p x n p. Since α n 0 as n, we have lim u n x n = 0. ISSN: 1109-2769 484 Issue 12, Volume 10, December 2011

From (11), we derive lim u n y n = 0. (13) Define T : C H by T x = λx + (1 λ)sx. Then T is nonexpansive with F (T ) = F (S) by Lemma 3. Since T u n u n T u n y n + y n u n λ β n u n Su n + y n u n, by using (13) and β n λ we obtain lim T u n u n = 0. Since {u n } is bounded, there exists a subsequence {u ni } which converges weakly to q. We shall show that q F (S) EP (ϕ). In fact, C is closed and convex, and hence C is weakly closed, we have q C. Let us show that q F (S). Assume that q F (T ), since u ni q and q T q, it follows from the Opial s condition that lim inf u n i q < lim inf u n i T q lim inf ( u n i T u ni + T u ni T q ) lim inf u n i q. This is a contraction. So we get q F (T ) and hence q F (S). Next, we show that q EP (ϕ). Since u n = T λn x n, for any y C we have y u n, u n x n 0. From (A2), it holds that 1 y u n, u n x n ϕ(y, u n ). Replacing n by n i, we have y u ni, u n i x ni ϕ(y, u ni ). i Since u n i x ni i 0 and u ni q, it follows from (A4) that 0 ϕ(y, q) for all y C. Let z t = ty + (1 t)q, t (0, 1], y C. Then we have z t C and hence ϕ(z t, q) 0. From (A1) and (A4) we get 0 = ϕ(z t, z t ) tϕ(z t, y) + (1 t)ϕ(z t, q) tϕ(z t, y), and hence 0 ϕ(z t, y). From (A3) we get 0 ϕ(q, y) for all y C and hence q EP (ϕ). Therefore, q EP (ϕ) F (S). On the other hand, since x n q = α n (µf γf)q + [I α n (µf γf)]y n [I α n (µf γf)]q we have x n q 2 = α n (µf γf)q, x n q + [I α n (µf γf)]y n [I α n (µf γf)]q, x n q α n (µf γf)q, x n q +[1 α n (τ γρ)] x n q 2, which follows that x n q 2 in particular, x ni q 2 1 τ γρ (µf γf)q, x n q, 1 τ γρ (µf γf)q, x n i q. Since x ni q, it follows from above that x ni q as i. Next, we show that q is a solution of the variational inequality (8). Since we have x n = [I α n (µf γf)]y n = [I α n (µf γf)]s n T λn x n, (µf γf)x n = 1 α n [(I S n T λn )x n α n (µf γf)(i S n T λn )x n ]. For any p EP (ϕ) F (S), (µf γf)x n, x n p = 1 α n (I S n T λn )x n α n (µf γf) (I S n T λn )x n, x n q = 1 α n (I S n T λn )x n (I S n T λn )q, x n q + (µf γf)(i S n T λn )x n, x n q. (14) Note that (I S n T λn ) is monotone (i.e., x y, (I S n T λn )x (I S n T λn )y 0, for all x, y H, due ISSN: 1109-2769 485 Issue 12, Volume 10, December 2011

to the nonexpansivity of S n T λn ). Replacing n in (14) by n i and letting i, we obtain (µf γf)q, q p = lim (µf γf)x ni, x ni p i lim (µf γf)(i S ni T λni )x ni, x ni p i = 0. (15) That is, q EP (ϕ) F (S) is a solution of (8). To show that the sequence {x n } converges strongly to q, we assume that x nk ˆx. Similar to the proof above, we have ˆx EP (ϕ) F (S). Moveover, it follows from the inequality (15) that (µf γf)q, q ˆx 0. (16) Interchange q and ˆx to obtain (µf γf)ˆx, ˆx q 0. (17) From Lemma 6, adding up (16) and (17) yields (µη γρ) q ˆx 2 q ˆx, (µf γf)q (µf γf)ˆx 0. Hence, q = ˆx, and therefore x n q, as n, [I (µf γf)]q q, q p 0 p EP (ϕ) F (S). This is equivalent to the fixed point equation P EP (ϕ) F (S) [I (µf γf)]q = q. The desired result follows. Theorem 8 Let C be a nonempty closed convex subset of a real Hilbert space H and ϕ be a bifunction from C C into R satisfying (A1), (A2), (A3) and (A4). Let S : C H be a k-strictly pseudo-contractive nonself mapping such that F (S) EP (ϕ). Let F : H H be an L-Lipschitzian continuous and η- strongly monotone operator on H with L, η > 0 and 0 < γ < τ ρ, τ = µ(η µl2 /2). Let {x n } be a sequence generated by: x 1 H arbitrarily, y n = β n u n + (1 β n )Su n, x n+1 = [I α n (µf γf)]y n, n N, where u n = T λn x n, y n = S n u n, if {α n }, {β n } and { }satisfy the following conditions: (i) {α n } (0, 1), lim α n =0, n=1 α n =, n=1 α n+1 α n < ; (ii) 0 k β n λ < 1 and lim β n = λ, n=1 β n+1 β n < ; (iii) { } (0, ), lim > 0, n=1 +1 < ; then {x n } and {u n } converges strongly to a point q F (S) EP (ϕ) which solves the variational inequality (8). Proof: We first show that {x n } is bounded. Indeed, pick any p F (S) EP (ϕ) to derive that x n+1 p = [I α n (µf γf)]y n p = (I α n µf )y n (I α n µf )p +α n γf(y n ) α n γf(p) + α n γf(p) α n µf p (1 α n τ) y n p +α n γρ y n p + α n (γf µf )p [1 α n (τ γρ)] x n p + α n (γf µf )p. By induction, we have { 1 } x n p max x 1 p, τ γρ (γf µf )p, and hence {x n } is bounded. From (9) and (10), we also derive that {u n } and {y n } are bounded. Next, we show that x n+1 x n 0. We have x n+1 x n = [I α n (µf γf)]y n [I α n 1 (µf γf)]y n 1 [1 α n (τ γρ)] y n y n 1 + α n α n 1 (γf µf )y n 1 [1 α n (τ γρ)] y n y n 1 + K α n α n 1, (18) where K = sup{ (γf µf )y n : n N} <. On the other hand, we have y n y n 1 = S n u n S n 1 u n 1 S n u n S n u n 1 + S n u n 1 S n 1 u n 1 u n u n 1 + S n u n 1 S n 1 u n 1. (19) From u n+1 = T λn+1 x n+1 and u n = T λn x n, we get ϕ(u n+1, y) + 1 +1 y u n+1, u n+1 x n+1 0 y C, (20) y u n, u n x n 0 y C. (21) ISSN: 1109-2769 486 Issue 12, Volume 10, December 2011

Putting y = u n in (20) and y = u n+1 in (21), we have ϕ(u n+1, u n ) + 1 +1 u n u n+1, u n+1 x n+1 0 y C, ϕ(u n, u n+1 ) + 1 u n+1 u n, u n x n 0 y C. From (A2) we get u n+1 u n, u n x n u n+1 x n+1 0, +1 and hence u n+1 u n, u n u n+1 + u n+1 x n +1 (u n+1 x n+1 ) 0. Since lim > 0, without loss of generality, we can assume that there exists a real number a such that > a > 0 for all n N. Thus, we have u n+1 u n 2 u n+1 u n, x n+1 x n +(1 λ n )(u n+1 x n+1 ) +1 { u n+1 u n x n+1 x n + (1 λ } n ) u n+1 x n+1, +1 and hence u n+1 u n x n+1 x n + +1 M 0 (22) a where M 0 = sup{ u n x n : n N}. Now we estimate S n u n 1 S n 1 u n 1. Notice that S n u n 1 S n 1 u n 1 = [β n u n 1 + (1 β n )Su n 1 ] [β n 1 u n 1 + (1 β n 1 )Su n 1 ] β n β n 1 u n 1 Su n 1. (23) From (22), (23) and (19), we obtain y n y n 1 x n x n 1 + M 0 a 1 + β n β n 1 u n 1 Su n 1 x n x n 1 + 1 M 1 + β n β n 1 M 1. (24) where M 1 is an appropriate constant such that M 1 M 0 a + u n 1 Su n 1, n N. From (18) and (24), we obtain x n+1 x n K α n α n 1 + (1 α n (τ γρ)) ( x n x n 1 + 1 M 1 + β n β n 1 M 1 ) (1 α n (τ γρ)) x n x n 1 +M( α n α n 1 + 1 + β n β n 1 ), where M=max[K, M 1 ]. By Lemma 5 we have lim x n+1 x n = 0. (25) Using (22) and (24) together with 1 0 and β n β n 1 0, we have From the equality it follows that lim u n+1 u n = 0. lim y n+1 y n = 0. x n+1 = [I α n (µf γf)]y n x n y n x n x n+1 + x n+1 y n = x n x n+1 +α n (γf µf )y n. From α n 0 and (25) we get lim x n y n = 0. (26) For p F (S) EP (ϕ), we have u n p 2 = T λn x n T λn p 2 which implies that x n p, u n p = 1 2 ( x n p 2 + u n p 2 u n x n 2 ), u n p 2 x n p 2 u n x n 2. (27) ISSN: 1109-2769 487 Issue 12, Volume 10, December 2011

Thus from (10) and (27) we derive that x n+1 p 2 = [I α n (µf γf)]y n p 2 = [I α n (µf γf)]y n [I α n (µf γf)]p α n (µf γf)p 2 [1 α n (τ γρ)] 2 y n p 2 +2α n (µf γf)p y n p +αn 2 (µf γf)p 2 u n p 2 +2α n (µf γf)p x n p +αn 2 (µf γf)p 2 x n p 2 u n x n 2 +2α n (µf γf)p x n p +α 2 n (µf γf)p 2. Since α n 0, x n x n+1 0, we have lim x n u n = 0. (28) From (26) and (28), we obtain u n y n u n x n + x n y n 0, as n. (29) Define a mapping T : C H by T x = λx+(1 λ)sx. Then T is nonexpansive with F (T ) = F (S) by Lemma 3. Since T u n u n T u n y n + y n u n λ β n u n Su n + y n u n, from (29) and β n λ we obtain lim T u n u n = 0. (30) Finally we show that lim sup (µf γf)q, q x n 0 where q = P F (S) EP (ϕ) [I (µf γf)]q is a unique solution of the variational inequality (8). Indeed, take a subsequence {x ni } of {x n } such that lim (µf γf)q, q x n i i = lim sup (µf γf)q, q x n. Due to {u ni } is bounded, there exist a subsequence {u nij } of {u ni } which converges weakly to w. Without lose of generality, we can assume that u ni w. From (28) and (30), we obtain x ni w and T u ni w. By the same argument as in the proof of Theorem 7, we have w EP (ϕ) F (S). Since q = P F (S) EP (ϕ) [I (µf γf)]q, it follows that From we get lim sup (µf γf)q, q x n = (µf γf)q, q w 0. (31) x n+1 q = α n (µf γf)q + [I α n (µf γf)]y n [I α n (µf γf)]q x n+1 q 2 [I α n (µf γf)]y n [I α n (µf γf)]q 2 +2α n (µf γf)q, x n+1 q [1 α n (τ γρ)] 2 x n q 2 +2α n (µf γf)q, x n+1 q. This implies that x n+1 q 2 [1 2α n (τ γρ) + (α n (τ γρ)) 2 ] x n q 2 + 2α n (µf γf)q, x n+1 q = [1 2α n (τ γρ)] x n q 2 +(α n (τ γρ)) 2 x n q 2 +2α n (µf γf)q, x n+1 q = [1 2α n (τ γρ)] x n q 2 [ αn (τ γρ) +2α n (τ γρ) M 2 + 1 ] τ γρ (µf γf)q, x n+1 q = (1 γ n ) x n q 2 + γ n δ n where M = sup{ x n q 2 : n N}, γ n = 2α n (τ γρ) and δ n = α n(τ γρ) 2 M + 1 τ γρ (µf γf)q, x n+1 q. It is easy to see that lim γ n = 0, n=1 γ n = and lim sup δ n 0 by (31). By Lemma 5, the sequence {x n } converges strongly to q. Remark 9 If C = H, S is a nonexpansive mapping, {β n } = 0, ϕ(x, y) = 0 for all x, y C, = 1, µ = 1 and F = A, then Theorem 8 reduced to Theorem 3.4 of G. Marino and H. K. Xu [6]. Remark 10 If µ = 1 and F = B in Theorem 7 and Theorem 8, we can obtain the corresponding results in Y. Liu [16]. Acknowledgements: The authors thank the anonymous referees for their careful reading, comments and suggestions to this paper. This research was supported by the Fundamental Research Funds for the Central Universities (grant No. ZXH2011C002 ). ISSN: 1109-2769 488 Issue 12, Volume 10, December 2011

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